1. Field of the Invention
The present invention relates to nuclear quadrupole resonance (NQR). More specifically, the present invention relates to the excitation of two frequencies to create the desired signal at a third frequency, and also the method and techniques for the creation of spin echoes and other signals analogous to those obtained with standard multipulse single-frequency NQR methods.
2. Description of the Related Art
Nuclear quadrupole resonance (NQR) is a technique for detecting target specimens containing sub-kilogram quantities of narcotics and explosives having quadrupolar nuclei. Such substances include nitrogenous or chlorine-containing explosives and narcotics. Basically, quadrupolar nuclei will exhibit nuclear quadrupole resonance—a change in the angle of nuclear spin with respect to its quantization axis when it is excited by radio frequency (RF) radiation pulses at a particular frequency. In the better-known nuclear magnetic resonance (NMR), the quantization axis is determined externally by the direction of the applied magnetic field. In NQR, the quantization axis is determined by molecular parameters. As with NMR, different chemicals require pulses of different frequencies (i.e., different nuclear quadrupole resonance frequencies) to cause precession in nuclei. A device used to detect magnetic or NQR resonance in the quadrupolar nuclei of a target specimen is tuned to emit pulses at the frequency corresponding to the resonance frequency of the nuclei desired to be detected. A typical NQR excitation/detection circuit consists of an inductor capacitively tuned to the NQR frequency and nominally matched to the impedance of a transmitter or receiver by another capacitor or inductor. In this regard, the present invention is related to other methods of NQR detection as taught in U.S. Pat. No. 5,206,592 issued Apr. 27, 1993 to Buess et al. for DETECTION OF EXPLOSIVES BY NUCLEAR QUADRUPOLE RESONANCE, and U.S. Pat. No. 5,233,300 issued Aug. 3, 1993 to Buess et al. for DETECTION OF EXPLOSIVE AND NARCOTICS BY LOW POWER LARGE SAMPLE VOLUME NUCLEAR QUADRUPOLE RESONANCE, and U.S. Pat. No. 5,365,171 issued Nov. 15, 1994 to Buess et al. for REMOVING THE EFFECTS OF ACOUSTIC RINGING AND REDUCING THE TEMPERATURE EFFECTS IN THE DETECTION OF EXPLOSIVES BY NQR, and U.S. Pat. No. 5,608,321 issued Mar. 4, 1997 to Garroway et al. for A MEANS FOR DETECTING EXPLOSIVES AND NARCOTICS BY STOCHASTIC NUCLEAR QUADRUPOLE RESONANCE (NQR), and U.S. Pat. No. 5,804,967 issued Sep. 8, 1998 to Miller et al. for A MEANS FOR GENERATING SHORT RF PULSES WITH RAPID DETECTOR RECOVERY IN STOCHASTIC MAGNETIC RESONANCE, and U.S. Pat. No. 6,242,918 issued Jun. 5, 2001 to Hepp et al. for APPARATUS AND METHOD FOR REDUCING THE RECOVERY PERIOD OF A PROBE IN PULSED NUCLEAR QUADRUPOLE RESONANCE AND NUCLEAR MAGNETIC RESONANCE DETECTION SYSTEMS BY VARYING THE IMPEDANCE OF A LOAD TO REDUCE TOTAL Q FACTOR, and U.S. Pat. No. 6,054,856 issued Apr. 25, 2000 to Suits et al. for MAGNETIC RESONANCE DETECTION COIL THAT IS IMMUNE TO ENVIRONMENTAL NOISE, and U.S. Pat. No. 6,104,190 issued Aug. 15, 2000 to Buess et al. for MEANS FOR DETECTING NITRAMINE EXPLOSIVES BY 14N NQR OF NITRO GROUPS, all of which are incorporated by reference herein.
It is common to detect a magnetic resonance signal by placing a sample to be measured in a tuned, electronically resonant tank circuit. Then, the response of the tank circuit to the electromotive force produced by nuclear or electronic spins in the sample is measured. With Nuclear Magnetic Resonance (NMR) or Nuclear Quadrupole Resonance (NQR), the sample is placed in or near an inductor, commonly referred to as a coil, that detects AC magnetic fields.
The inductance of the coil is tuned with a parallel and/or series capacitance to make the circuit electrically resonant at the measurement frequency. One or more additional reactive impedances (inductors or capacitors) are typically added to adjust the resistive impedance at resonance to a particular value which optimizes the detection sensitivity.
Although the NQR detection technique works reasonably well in some circumstances, one of the challenges that NQR faces is that when applying the RF magnetic field needed to detect NQR, one can create false signals at or near the NQR frequency. For example, acoustic vibrations are created in certain magnetic metals. In turn, as the magnetic domains in the metal vibrate, they induce a signal back in the NQR receiver coil at essentially the same frequency as the driving frequency. For example, if one has a suitcase with chrome trimmings, the detected signal may indicate that there are explosives in the suitcase when in fact it is acoustic ringing from the chrome that is observed.
A multiple-frequency technique would eliminate the false alarms due to acoustic ringing. For example, for a spin-1 nucleus such as 14N, the three transition frequencies between the levels are discrete. The basic concept is to use two of the frequencies to excite the third transition, and to then detect the signal from this third transition. This avoids any interference from acoustic ringing since the RF is not applied at the frequency that is detected. The method disclosed here is the first direct observation of a NQR transition near a frequency that has not been used for irradiation.
There is another advantage to not irradiating at the observation frequency: the probe recovery time is greatly reduced. Following a high power RF pulse one has to wait for the energy stored in the coil to dissipate before signal can be detected. This recovery time is typically about 20 coil ringdown time constants. The ringdown time is proportional to the coil quality factor, Q, and inversely proportional to the frequency. At low frequency, a substantial portion of the signal may be lost during the recovery time. In three-frequency NQR, the energy stored in the observation coil is limited to that which may leak in from the excitation at other frequencies. Therefore, improved sensitivity is expected with three-frequency NQR at low frequencies when the signal lifetime is short.
The nuclear wave function evolves under a Hamiltonian consisting of the large time-independent quadrupole term and the much smaller time-dependent terms corresponding to the alternating RF magnetic fields applied at two of the three characteristic NQR frequencies. For example, with frequencies for spin-1 nuclei and the principal axes frame (x, y, z) of the electric field gradient tensor at the quadrupolar nucleus, the quadrupolar Hamiltonian HQ is HQ=e2qQ[(3Iz2−I2)+η(Ix2−Iy2)]/4 where I is the nuclear angular momentum operator, η the asymmetry parameter of the electric field gradient, q the field gradient, and Q the quadrupole moment of the nucleus. The transition frequencies between the eigenfunctions of HQ are ω±=(3±η)e2qQ/4 ℏ and ω0=ηe2qQ/2 ℏ. Here, it is assumed η≠0 or 1 to avoid degenerate energy levels and transition frequencies which complicate the calculation. To be concise, only the observation at ω+ arising from irradiation at ω− and ω0 is here treated in detail, but similar results are expected for other three-frequency combinations.
For a spin-1 nucleus each transition is allowed under a different orientation of the applied field in the principal axes frame (i.e., <+|I|−>=<+|Iz|−>, <0|I|−>=<0|Iy|−>, and <+|I|0>=<+|Ix|0>). Therefore, for a single crystal, the most efficient NQR excitation and detection occur when an RF magnetic field of frequency ω0 is applied along the z-direction, that of ω− along y, and the ω+ detection coils are sensitive to magnetization oscillating along x. Similarly, for a powder sample it can be shown that the maximum signal is obtained if the two RF magnetic fields are mutually perpendicular to one another in the laboratory frame (x′, y′, z′). The received ‘three-frequency’ NQR signal then arises from a magnetization which is orthogonal to both the applied RF fields.
The Hamiltonian for the interaction of the nucleus with an RF pulse of magnetic field strength B1− and frequency ω− applied along the x′-axis is H1−=−ℏγNB1−Ix′ cos ω−t and with an RF pulse of strength B10 and frequency ω0 along y′ is H10=−ℏγNB10Iy′cos ω0t. (γN is the gyromagnetic ratio of the nucleus.) The lab frame operators can be expressed asIx′=(cos α cos β cos γ−sin α sin γ)Ix+(sin α cos β cos γ+cos α sin γ)Iy−sin β cos γIz Iy′−(−cos α cos β sin γ−sin α cos γ)Ix+(−sin α cos β sin γ+cos α cos γ)Iy+sin β sin γIz  (1)where α, β, and γ are Euler angles describing the relative orientation of the principal axes and lab frames. If it is assumed each RF pulse only excites one transition, H10 and H1− can be simplified asH1−=−ℏγNB1−(sin α cos β cos γ+cos α sin γ)Iy cos ω−t≡−ℏΩ1−Iy cos ω−t H10=−ℏγNB10(sin β sin γ)Iz cos ω0t≡−ℏΩ10 Iz cos ω0t  (2)where the dependence on crystal orientation is now contained implicitly in the newly defined terms Ω10 and Ω1−, the effective RF-nutation rates.
Using the above Hamiltonians, the wave function |Ψ(t)> is found after a single RF pulse of duration tp at a frequency ω0, can be written as a simple rotation of the original wave function around z, |Ψ(t)>=e−iHQt/ℏeilzθ|Ψ(0)>, where θ=Ω10tp/2. Similarly an RF pulse at the frequency ω− is equivalent to a rotation about y by Ω1−tp/2. Furthermore, the wave function after simultaneous irradiation at ω− and ω0 can be shown to be equivalent to a rotation about an axis in the y-z plane, rotated from the z-axis by an angle ξ, where tan ξ=Ω1−/Ω10. That is |Ψ(t)>=e−iHQt/ℏet(cos ξlz+sin ξIy)θ|Ψ(0)>, where θ=√{square root over ((Ω102+Ω1−2))}tp/2 is the angle of rotation defined by the effective RF-field generated by the two orthogonal RF fields. For notational simplicity, the above cases assume that all pulses are referenced to zero phase. The effect of including non-zero phases is simply the addition of the phases of the ω− pulse and the ω0 pulse to the final phase of the signal.
Consider serial irradiation, where irradiation at the two frequencies occurs at different times. Using the above operators and starting from thermal equilibrium, it is found that a pulse of length tpa at ω− followed by a pulse of length tpb at ω0 with a delay of τ between the pulses results in an expectation value oscillating at ω+ given by                               〈                                    I              x                        ⁡                          (              t              )                                〉                =                              (                                          N                0                0                            -                              N                -                0                                      )                    ⁢                      sin            ⁡                          (                                                Ω                                      1                    -                                                  ⁢                                  t                  p                  a                                            )                                ⁢                      sin            ⁡                          (                                                                    Ω                    10                                    ⁢                                      t                    p                    b                                                  2                            )                                ⁢                      sin            ⁡                          (                                                                    ω                    +                                    ⁡                                      (                                          t                      +                                              t                        p                        b                                                              )                                                  +                                                      ω                    -                                    ⁡                                      (                                          τ                      +                                              t                        p                        a                                                              )                                                              )                                                          (        3        )            where t is the time after the end of the second pulse. N−0 and N00 are the thermal populations of the eigenstates |−> and |0>, so that the amplitude of the signal depends on the initial difference in the populations connected by the first transition excited. Note that for a single crystal the maximum signal occurs when the first pulse induces a nutation angle of Ω1−tpa=π/2 and the second pulse has Ω10tpb=π, or for a crystal oriented for the most efficient excitation γNB1−tpa=π/2 and γNB10tpb=π. For a powder, the observed signal is proportional to the average over all possible crystal directions of the nuclear spin angular momentum projected along the axis of the detection coil. Usually no signal at ω+ is observed using receiver coils oriented in the x′-y′ plane. With a detection coil along the z′-axis the signal is proportional to                               S                      z            ′                          ∝                              ∫            0                          2              ⁢              π                                ⁢                                           ⁢                                    ⅆ              α                        ⁢                                          ∫                0                π                            ⁢                                                           ⁢                                                ⅆ                  β                                ⁢                                                                   ⁢                sin                ⁢                                                                   ⁢                β                ⁢                                                      ∫                    0                                          2                      ⁢                      π                                                        ⁢                                                                           ⁢                                                            ⅆ                                              γ                        ⁡                                                  (                                                      cos                            ⁢                                                                                                                   ⁢                            α                            ⁢                                                                                                                   ⁢                            sin                            ⁢                                                                                                                   ⁢                            β                            ⁢                                                                                          ⅆ                                                                  〈                                                                                                            I                                      x                                                                        ⁡                                                                          (                                                                              α                                        ,                                        β                                        ,                                        γ                                                                            )                                                                                                        〉                                                                                                                            ⅆ                                t                                                                                                              )                                                                                      .                                                                                                          (        4        )            
Performing the powder average numerically, it is found that the maximum attainable signal occurs when γNB1−tpa=2.13 rad and γNB10tpb=4.26 rad rather than π/2 and π (the aligned single crystal results) because randomly oriented crystals will experience RF pulses reduced by the angular factors in Ω1− and Ω10 (see Eq. 2). This behavior mimics that seen in single frequency NQR where the maximum signal for a powder occurs at a nutation angle of 2.08 rad, a third longer than the nutation angle needed (π/2) for a properly oriented single crystal. For a powder, the three-frequency maximum signal size is             2      ⁢              (                  1          -                      η            /            3                          )                    3      ⁢              (                  1          +                      η            /            3                          )              =            2      ⁢              ω        -                    3      ⁢              ω        +            of the maximum signal of a single-frequency NQR experiment at ω+.
Similarly, simultaneous irradiation of the sample at ω− and at ω0 results in an expectation value oscillating at ω+ such that<Ix(t))=sin 2ξ(1−cos θ)sin(ω+t)×[(N00−N+0)(cos2ξ+sin2ξ cos θ)−(N−0−N+0)(1+cos θ)].  (5)
From examination of the geometrical terms, a maximum signal for a single crystal occurs when ξ=π/8 and θ=π, or for a crystal oriented for the most efficient excitation B1−/B10=tan(π/8) and γNB10tp=2π cos(π/8). Using <Ix(t)> of Eqn. 5, numerically integrate to find the powder-averaged signal in the detection coil oriented along z′. Again no signal at ω+ is observed using receiver coils oriented in the x′-y′ plane. Although the signal is dependent in a complicated manner on the thermal populations, and therefore η, the parameters which give a maximum signal depend only slightly on η(B1−/B10≈tan(π/8) and γNB10tp≈7.6 rad, approximately a third longer than 2π cos(π/8)). The maximum signal ranges from 67% to 53% (0<72<1) of the corresponding maximum signal for a single-frequency experiment at ω+.
In practice, there is a small distribution of quadrupole interactions within the sample so that HQ=HQ0+ΔHQ. A second simultaneous pulse applied at time τ after an initial simultaneous pulse can undo the dephasing caused by this distribution so that a spin-echo is formed at a time t=τ after the end of the second pulse (see FIG. 5). Assuming ΔHQ is small enough that it has a negligible effect on the time evolution during applied pulses (i.e., ΔHQ<<H1−, H10), it is found |Ψ(t)>=e−iHQ0(t+tpbτ+tpa)/ℏe−iΔHQt/ℏeiKbe−iΔHQτ/ℏeiKa|Ψ(0)>, where Ka=(cos ξaIz+sin ξaIy) θa corresponds to the first simultaneous pulse of length tpa and Kb=(cos ξbIz+sin ξbIy) θb to the second pulse of length tpb. The refocused signal at ω+ is then<Ix(t)>=−sin 2μa(1−cos θa)sin(ω+(t+tpb+τ+tpa)+Δω+(t−τ)+2Δφ)×[(N00−N+0)(cos2ξa+sin2ξa cos θa)−(N−0−N+0)(1+cos θa)]×sin2ξb cos2ξb(1−cos θb)2  (6)where Δω+ describes the distribution in ω+ and Δφ≡Δφ+Δφ0. (Δφ is the phase difference between the first and second pulse of ω− and Δφ0 the phase difference for the ω0 pulses.) For a single crystal, the distribution due to ΔHQ is completely refocused and the largest echo occurs when ξa=π/8, ξb=π/4, and θa=θb=π (or for a crystal oriented for the most efficient excitation γNBa10tpa=2π cos(π/8), γNBb10tpb=2π cos(π/4)). Through numerical integration, it is found that for a powder the echo is a maximum when Ba1−/Ba10≈tan(π/8), Bb1−/Bb10≈tan(π/4), γNBa10tpa≈7.4 rad, and γNBb10tpb=5.7 rad. The maximum signal ranges from 48% to 41% (0<η<1) of the corresponding maximum signal for a single-frequency resonant experiment at ω+, or approximately 75% of the signal following the first pulse is refocused for a powder sample. When deriving Eqn. (6) there are several terms not included that depend on the precise nature of the quadrupole field distribution and may give rise to echoes at times other than at t=τ, as discussed by Grechishkin.